Around the Dartboard

You may have noticed that the numbers on a standard dartboard have the
following order, going clockwise starting at the top: 20, 1, 18, 4, 13, 6, 10,
15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12, and 5.

Why this particular order?

A possible criterion for designing a dartboard is to penalize poor
shots as much as possible. That can be done, for example, by maximizing
the sum of the absolute value of the difference between adjacent
numbers. The larger this sum, the more a poor shot is penalized.

To achieve this maximum, the numbers 11 to 20 should interlace 1 to
10. In this case, the differences between adjacent numbers total 200.

Surprisingly, even though there are so many possible arrangements
that give the maximum penalty score of 200, the standard dartboard in
everyday use has a difficulty of only 198. The "flaw" lies in
the placement of 11 next to 14 and 6 next to 10. You could correct this
imperfection simply by inserting 14 between 6 and 10 to achieve the
maximum difficulty.

Of course, the sum of the differences between adjacent numbers represents just
one possible way of defining the difficulty of a dartboard. You might consider,
for example, the squares of the differences of adjacent sectors or not only
immediate neighbors but also near neighbors.

Moreover, the dartboard also includes areas representing doubles, trebles,
and the inner and outer bull. These features, which play an important role in
many games, considerably complicate any analysis of the board.

Expert players often concentrate on the treble 20 when aiming for a high
score. Players who throw with a much lower accuracy may be wise to aim at sector
14, which has 9 and 11 as adjacent sections. That's the area of the standard
dartboard that happens to deviate from the requirements for maximum difficulty,
at least according to one criterion.

It would be interesting to delve into the history of dartboards to see how
the standard numbering scheme came about.

Ivars Peterson is the mathematics and physics writer and online editor at Science
News ( http://www.sciencenews.org/).
He is the author of The Mathematical Tourist, Islands of Truth, Newton's
Clock, and Fatal Defect. His latest book, The Jungles of
Randomness: A Mathematical Safari, is to be published in October by Wiley.